Can I specify the direction along which Edge Slide follows?
I think what you are attempting can't be done. The direction of the slide isn't following any edge which is why it snaps to the direction that has an edge to actually follow. That suggested a workaround: extrude and scale (shift-Z) to create edges for the slide to follow. I just tried it to test and it allowed numerical entry after setting the slide direction. An easier solution is shown below. Simply snap the cursor to the pivot point you require and scale from the cursor. Hitting S->shift_Z->numerical value: